Cartan Subgroups from Maximal Tori
Let \(G\) be a connected linear algebraic group over an algebraically closed field, and let \(T\subseteq G\) be a maximal torus. Then \[ C_G(T)^\circ \] is a Cartan subgroup of \(G\). Conversely, every Cartan subgroup of \(G\) is of this form for a maximal torus \(T\). In particular, Cartan subgroups are conjugate.
If \(G\) is reductive, then \(C_G(T)=T\) for every maximal torus \(T\).
Proof
The centralizer \(C_G(T)\) is closed, and its identity component is connected and nilpotent in the classical algebraic-group sense. Maximality of \(T\) forces any semisimple part in the centralizer to lie in \(T\), so \(C_G(T)^\circ\) is the largest connected subgroup whose semisimple part is controlled by \(T\). This is exactly the Cartan condition.
Conversely, a Cartan subgroup contains a maximal torus, and its defining self-normalizing nilpotence forces it to coincide with the connected centralizer of that torus. Since maximal tori are conjugate, the resulting Cartan subgroups are conjugate. In a reductive group, a nontrivial unipotent part in \(C_G(T)^\circ\) would define a root-zero unipotent normal direction, contradicting reductivity; hence the centralizer is the torus itself.